Theorem sslin 3359

Description: Add left intersection to subclass relation. (Contributed by NM, 19-Oct-1999.)

Assertion

Ref	Expression
sslin	⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∩ 𝐴) ⊆ (𝐶 ∩ 𝐵))

Proof of Theorem sslin

Step	Hyp	Ref	Expression
1		ssrin 3358	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶))
2		incom 3325	. 2 ⊢ (𝐶 ∩ 𝐴) = (𝐴 ∩ 𝐶)
3		incom 3325	. 2 ⊢ (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶)
4	1, 2, 3	3sstr4g 3196	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∩ 𝐴) ⊆ (𝐶 ∩ 𝐵))

Syntax hints:   → wi 4   ∩ cin 3126   ⊆ wss 3127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-in 3133  df-ss 3140

This theorem is referenced by:  ss2in  3361  difdifdirss  3505  ssres2  4927  ssrnres  5063  sbthlem7  6952  ioodisj  9964  ntrss  13190  cnptoprest  13310